As you did in previous assignments, make a population of 20 LIF neurons representing a 1-dimensional value, and compute a decoder for them. For parameters, $\tau_{ref}$=0.002s, $\tau_{RC}$=0.02s, the maximum firing rates are chosen randomly from a uniform distribution between 100 and 200Hz (at the max radius), and the x-intercepts are chosen randomly from a uniform distribution between -2 and 2. Remember that the $\alpha$ and $J^{bias}$ terms are computed based on these x-intercepts and maximum firing rates.

It is generally easiest to compute decoders using the original method from Assignment 1, where we use the rate-mode approximation for the neurons to generate the $A$ matrix, then find $\Gamma=A^T A + \sigma^2 I$. You can use this approach to find decoders, and these decoders should work even when you simulate the neurons in terms of spikes (in question 2 on). The only difference will be that they will need to be scaled by dt, your simulation time step.

Use this same method for computing decoders for this whole assignment.

[0.5 marks] Plot the tuning curves (firing rate of each neuron for different $x$ values between -2 and 2)

[0.5 marks] Compute the decoders and plot $(x-\hat{x})$. When computing decoders, take into account noise ($\sigma$=0.1 times 200Hz). When computing $\hat{x}$, add random gaussian noise with $\sigma$=0.1 times 200Hz to the activity. Report the Root Mean-Squared Error (RMSE).

Choose a neuron from part 1 that has a firing rate of somewhere between 20-50Hz for $x$=0. Using that neuron's $\alpha$ and $J^{bias}$ value, construct two neurons: both with the same $\alpha$ and $J^{bias}$, but one with $e$=+1 and the other with $e$=-1. With the function from the last assignment, generate a random input $x(t)$ that is 1 second long, with rms=1, dt=0.001, and an upper limit of 5Hz. Feed that signal into the two neurons and generate spikes. Decode the spikes back into $\hat{x}(t)$ using a post-synaptic current filter $h(t)$ with a time constant of $\tau$=0.005.

Repeat question 2, but with more neurons. Instead of picking particular neurons, randomly generate them with x-intercepts uniformly distributed between -2 and 2 and with maximum firing rates between 100 and 200 Hz. Randomly choose encoder values to be either -1 or +1.

[2 marks] Plot Root Mean-Squared Error as the number of neurons increases, on a log plot. Try 8 neurons, 16 neurons, 32, 64, 128, up to 256. For the RMSE for a particular number of neurons, average over at least 5 randomly generated groups of neurons. For each group of neurons, randomly generate the signal $x(t)$. Use the same parameters as in question 2. Note: the RMSE should go down as the number of neurons increases

For this question, use two groups of neurons with intercepts between [-1, 1] (i.e. radius = 1) to compute $y = 2x+1$. The first group of neurons will represent $x$ and the second group will represent $y$.

Start by computing decoders. You will need two decoders: one to decode $f(x)=2x+1$ from the first population, and one to decode $f(y)=y$ (the standard representational decoder) from the second population. Remember that $\Upsilon$ can change depending on what function you want to decode.

Use the same neuron parameters as for previous questions (other than the radius), and use 200 randomly generated neurons in each population.

[1 mark] Show the behaviour of the system with an input of $x(t)=t-1$ for 1 second (a linear ramp from -1 to 0). Plot the ideal $x(t)$ and $y(t)$ values, along with $\hat{y}(t)$.

Note that you should use the decoders that work for any input over the range of intercepts: do not re-compute decoders for any particular input (i.e. set of $x$ values).

Input $x(t)$ into the first group of neurons and produce spikes. Decode from those spikes using the decoder for $f(x)=2x+1$. Input that decoded result into the second group of neurons to produce spikes. Use the second decoder ($f(y)=y$) to decode $\hat{y}(t)$.

Make sure the maximum firing rates are now at -1 or 1 (i.e., the radius is 1).

[0.5 marks] Repeat part (a) with an input that is ten randomly chosen values between -1 and 0, each one held for 0.1 seconds (a randomly varying step input)

[0.5 marks] Repeat part (a) with an input that is $x(t)=0.2sin(6\pi t)$. Briefly discuss the results for this question.

For this question, use three groups of neurons with intercepts from [-1, 1] to compute $z = 2y+0.5x$. Follow the same steps as question 4, but take the decoded outputs from the first two groups of neurons ($f(y)=2y$ and $f(x)=0.5x$), add them together, and feed that into the third group of neurons.

[0.5 marks] Plot $x(t)$, $y(t)$, the ideal $z(t)$, and the decoded $\hat{z}(t)$ for a random input over 1 second. For $x(t)$ use a random signal with a limit of 8 Hz and `rms`=1. For $y(t)$ use a random signal with a limit of 5 Hz and `rms`=0.5.

Do the same thing as questions 4 and 5, but with 2-dimensional vectors instead of scalars. Everything else is the same. For your encoders $e$, randomly generate them over the unit circle.

The function to compute is $w = x-3y+2z-2q$. This requires five groups of neurons: $x$, $y$, $z$, $q$, and $w$. Each of them represents a 2-dimensional value. The outputs from $x$, $y$, $z$, and $q$ all feed into $w$.

[1 mark] Plot the decoded output $\hat{w}(t)$ and the ideal $w$ for $x=[0.5,1], y=[0.1,0.3], z=[0.2,0.1], q = [0.4,-0.2]$. (Note that these are all constants so they don't change over time, but still plot it for 1.0 seconds on one or more 2D graphs)